Problem 25 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} y>x \\ y>2 \end{array}\right) $$

Step-by-Step Solution

Answer

Shade the region above both \( y = x \) and \( y = 2 \).

Step 1: Understand the inequalities

The given system consists of two inequalities: \( y > x \) and \( y > 2 \). These inequalities describe a region in the coordinate plane where the values of \( y \) are greater than those of \( x \) and also greater than 2.

Step 2: Graph the inequality \( y > x \)

To graph \( y > x \), first graph the line \( y = x \), which is a line through the origin with a slope of 1. Since the inequality is \( y > x \), shade the region above this line, where the y-values are greater.

Step 3: Graph the inequality \( y > 2 \)

Next, graph the line \( y = 2 \), which is a horizontal line intersecting the y-axis at 2. The inequality \( y > 2 \) indicates that we should shade the region above this line, where the values of \( y \) are greater than 2.

Step 4: Determine the solution set

The solution set is the region that satisfies both inequalities. This is where the shading from \( y > x \) and \( y > 2 \) overlap. Hence, the solution set is the area above both the line \( y = x \) and the line \( y = 2 \).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and regions, based on their coordinates. These coordinates are labeled as

• a horizontal axis (commonly known as the x-axis)
• a vertical axis (known as the y-axis).

Each point on this plane is represented by an ordered pair \((x, y)\), where \('x'\) is the x-coordinate and \('y'\)\ is the y-coordinate. The intersection of these axes is called the origin, represented as \( (0, 0) \).
The plane is divided into four quadrants:

• Quadrant I: both x and y are positive,
• Quadrant II: x is negative, and y is positive,
• Quadrant III: both x and y are negative,
• Quadrant IV: x is positive and y is negative.

Whenever solving inequalities, the coordinate plane helps determine which areas satisfy the given conditions by allowing visual representation of those inequalities as shaded areas. By plotting lines and shading specific regions, we can easily see where inequalities overlap and visualize their common solution area.

Graphing Inequalities

Graphing inequalities involves plotting lines or curves on the coordinate plane to show which regions satisfy the inequality conditions.For example, in our exercise, we need to graph the inequalities \(y > x\) and \(y > 2\).
Let's break this down step by step:

• Start by graphing the line of each inequality boundary. For \(y > x\), first graph the line \(y = x\), which passes through the origin and has a slope of 1. For \(y > 2\), graph the horizontal line \(y = 2\) across the y-axis.
• These lines divide the plane into different regions. We need to determine which of these regions satisfy the inequalities.
• Both lines should be represented as dashed lines on the graph since the solutions do not include the line itself (because we have \(>\), not \(\geq\)).

By graphing these inequalities on the coordinate plane, it becomes easier to understand where these conditions are fulfilled.

Shading Regions

Shading regions on a graph is a crucial step in solving systems of inequalities. It visually highlights the parts of the coordinate plane where solutions to the inequalities exist.
For the system \(y > x\) and \(y > 2\), follow these shading steps:

• After graphing \(y = x\), shade the region above this line, because the inequality is \(y > x\). This suggests all points where the y-coordinate is greater than the corresponding x-coordinate.
• After graphing \(y = 2\), shade the region above the line. This highlights all the points where the y-coordinate is greater than 2.
• The solution set to the system is found where these two shaded areas overlap. This overlapping region satisfies both inequalities simultaneously.

By clearly shading these areas, you easily identify the solutions that work for both inequalities, providing a visual solution to the problem that complements the algebraic approach.